 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2012/s25/1/3/b/solution

For the quadratic Planck function profile,

I_{0} = π B_{0} + 2 π β \int_{0}^{τ_{s}} τ e^{- τ} d τ = π B_{0} + 2 π β [1 - (1 + τ_{s}) e^{- τ_{s}}] .

The optically thick limit and the instantaneous doubled-abundance result are therefore

I_{0} = π (B_{0} + 2 β), I_{0} = π (B_{0} + β /2), Δ I_{0} = - \frac{3}{2} π β .

The new coefficient is

β /4

, since

B (τ) = B_{0} + β (τ /2)^{2}

at fixed physical temperature. For positive

β

, the outgoing spectral irradiance decreases. The factor differs from the linear case because the temperature-dependent source function curves with depth; doubling abundance does not mean holding its old functional coefficients fixed in the new optical-depth coordinate.

 New to topics? Read the docs here!